Conditional Preference Orders and their Numerical Representations

TL;DR

This paper develops an axiomatic framework for modeling conditional preferences using conditional set theory, introduces numerical representations, and extends classical theorems like Debreu's and von Neumann-Morgenstern's to the conditional setting.

Contribution

It introduces a novel axiomatic system for conditional preferences and extends key representation theorems to the conditional context, using a new conditional axiom of choice.

Findings

Conditional numerical representations are established.

Conditional versions of Debreu's and von Neumann-Morgenstern theorems are proved.

Conditional continuity results are demonstrated with examples.

Abstract

We provide an axiomatic system modeling conditional preference orders which is based on conditional set theory. Conditional numerical representations are introduced, and a conditional version of the theorems of Debreu on the existence of numerical representations is proved. The conditionally continuous representations follow from a conditional version of Debreu's Gap Lemma the proof of which relies on a conditional version of the axiom of choice, free of any measurable selection argument. We give a conditional version of the von Neumann and Morgenstern representation as well as automatic conditional continuity results, and illustrate them by examples.